Jonathan and Peter Borwein are Canadian mathematicians
being a part of the CECM connected with the university Simon Fraser of Vancouver.Jonathan is the director and his brother Peter is the associated director.
For more information, click their names and you will go on their personal page.

About

For more than 15 years, these two revolutionized
research on Pi! After the algorithm with quadratic convergence found by Brent/Salamin in 1976, they practically then monopolized the discoveries of series.
Quadratic, cubic, quadric, nonic the speed of convergence did not stop since!
In fact, they proved some years ago, that an algorithm with speed n-ic converging
Pi exists for every integer n... But the complexity of calculations increase very
fast and it seems that the algorithm with quadric convergence represents the best
ratio complexity/speed... It was moreover used in most of records since its discovery
and notably by Kanada to calculate 51 billions of decimal recently.
You will have understood it, Borwein represents with the small group composed of Chudnovsky, Simon Plouffe, Garvan, Gosper et Bailey, the highlight of the active
research on Pi today.

As
regards the proof of these formulae, I must regrettably pass besides the great principle
of this site... Because these proofs are already transcribed on Web at the following
address: www.cecm.sfu.ca/organics/papers/garvan/paper/html/paper.html. It is necessary to recognize also that it would be useless
to copy out silly 4 or 5 summaries of demonstration without being able to complete
the intermediaries of calculation (even though principle is rather simple to understand),
because not being of the level of a mathematician...

I retranscribe so only the summary of the proof for the 3) (2-nd formula), which
is perfectly representative of the principle of demonstrations using modular equations
(to see the page dedicated to Ramanujan for the explanation
of this theorie and of the principle of the démonstration)

Proof

The techniques of this proof mainly appeal to theta
functions, to the Eta function of Dedekind and to their properties.

1.1 Introduction :

So let us put Thêta functions:, r>0

Then, function from the Borwein is introduced : rR+*,

When r->+, q -> 0 and .

We are going to build an infinity of series p and find a relation between p(N2r) and p(r) (As the Borwein say, it is a "nice" relation
if N=p !)

2.6 and
according to 2.2,
We have then
(All these previous relations are very difficult to find but fortunately true (!)
and principle is interesting)

2.7 So
for r=1 we have (Does not depend of !!)
2.8 Besides
we have, as well as for :
3.1 Relation
between p(N2r) and p(r)

Let N, p1, we obtain
:

Exceptional relation on which are based all these algorithms!!

3.2 According
to 2.4,
we have :

(Used series is called the series of Eisenstein E2(q))

3.3 According
to 3.1,
we still have :

3.4 And
so for N=p we have according to 3.1 :

Now according to 3.1 and 2.6 we have :

mp,p()=p

3.5 So we
find for r= and according to 2.6 :

p()=

(Well, would not it be a good point of departure
for =0,318... !!)

4.1 Construction of the algorithm

We put

n=p(r0 p2n)

r0 p2n -> + when n -> +
so n ~ because p(r0 p2n) -> according to 2.8.
We also put

mn=mp,p(r0 p2n)

This writing not being what we can call the most
practical, so we are going to find a relation between n and n-1 knowing that :0=p(r0) for r0=, so 0=.

The first part of the general study is ended. To build the algorithms, the series
consists in choosing one singular p and then to introduce modular forms a(q), b(q) and c(q). The existence
of a relation between a(q), b(q), c(q) and a(qP ), b
(qP ), c(qP) supplies
modular equation

a p+b p=c p

(No, no, it is not Last Fermat's theorem !!)

By defining :, we
have sp+(s*) p=1 and relations between s, s* and supply the algorithm...

Application to the order 2 (p=2) :

5.1 Definitions :

According to 3.2,
we have A2(q)=2P(q2)-P(q)=34(q)+24(q) according to 1.1 and 3.2

Let us put besides :

5.2 a(q)=34(q)+24(q)
5.3 b(q)=44(q)
5.4 c(q)=222(q)32(q)

Associated modular equation is :

5.5 a2=b2+c2

With 3.1 we have :

5.6
Then according to 5.2 à 5.5, we find (not easily) :

5.7 and
5.8 a(q)=a(q2)+3c(q2)

By posing s and s* as defined higher, we deduct from it :
5.9 and
5.10 ,
good...

We have with 5.5 :
5.11 s2+(s*)2=1

According to 5.9 and 5.6,
we have :
5.12 m2,2(r)=(1+3s(q2)) (not too difficult, that!)

And, with 5.10 and 5.9

On the other hand, according to 3.4 and 5.12, we have

5.13
Now, of 3.4 and 3.5,
we know that 2()= and m2,2()=p=2
and according to 5.12, 2=1+3S(2) so S(2)=.
5.14 One has then n=(4n)
5.15 and sn=S(4n)
So, finally, we have 0=, s1=S(2)=,
and
and with 5.13,

n=(1+3sn)n-1 - 2n-1sn

Where from we obtain (finally!) the algorithm :

That is the most frustrating in this proof, it
is the simplicity of the principle and the parallel horror of the calculations that
does not appear here because the most difficult results are admitted...

Concerning series, I have the general formula of forming these series and the principle,
but not having understood everything, I let you to go to watch the site of Borwein
:

First series corresponds to the case t=427 and the
second in the case t=1555.

Trials

No attempts found on the net, so, there here is.
Obviously, they are quite amazing!
The second formula was tested also in the chapter Salamin...

For the following ones, let us cling! Order indicates the speed of
convergence (2-> quadratic, 4-> quadric), and the figure in brackets formula
to that test refers...

n=

1

2

3

4

5

6

7

8

9

10

12

Ordre 2 (2)

2

8

18

40

83

170

344

693

1392

2789

Ordre 2 (3.2)

0

2

7

18

40

84

171

345

694

Ordre 4

7

40

170

694

29360128

Ordre 5

5

31

166

848

I was not able to test upper orders in 5, my exceeding current computer 100 decimal
of preciseness (that is pitifully small, on looking at the speed of the convergence!)